In the ARAM queue that arrived with the Freljord patch you can re-roll a champion you've been given by using a re-roll credit. How fast do you accrue these re-rolls? Is it based on number of games played or number of champions owned?
You gain Reroll points based on games played and champions owned. Once you've saved up 250 Reroll points, it grants you 1 Reroll, which is used to perform a Reroll in ARAM. You can only have 2 Rerolls saved up; When you have 2 Rerolls, you no cannot accrue Reroll points.
The formula for Reroll points gained is
30 + (1 x Champions Owned) per game. This has been stated in the forums by a developer, and is very easy to confirm in the post-game stats by mousing over the Reroll point awards.
Standard 30 Points
Bonus From Champions Owned
Note that I own 58 champions.
From the Patch Notes:
Players may reroll for a different champion in Champion Select for 200 reroll points.
All players have been credited with 200 reroll points.
Additional reroll points are gained by completing matchmade Howling Abyss games.
Reroll point gains increase based on the number of champions the player owns.
Reroll point count can be checked in the player profile.
The only mention of the formula that I've come across is that a finished game nets you 30+Champions points, from here but I'm not sure how reliable it is.